Regularity for partial differential equations

本项目主要研究方向为非线性偏微分方程、变分学和奇异摄动理论。研究课题来自三种应用场景：超导的Ginzburg-Landau型模型、薄膜起泡的von Karman型模型和微磁学模型。在超导领域，P.I.将关注样品对大磁场的响应，特别关注从正常状态到超导状态的分岔。在薄膜水疱领域，P.I.将通过分析水疱生长和薄膜生长的各种动力学模型来研究水疱区域不稳定性的本质。最后，在微磁学领域，P.I.将分析探索一种模型，该模型被认为可以捕获与样品内几何收缩相关的新型磁壁结构。这个项目关注的是各种材料在受到外部环境或被迫呈现特定形状时的行为。这些系统的能量通常是通过一个函数来描述的，这个函数通常被称为“序参数”，它的值表明了在给定的一组环境下（如几何形状、应用场等）材料所呈现的状态。通过这种类型的研究，人们希望获得对给定样品的最佳形状的理解，以增强或减少各种物理效应。例如，在超导体的情况下，人们希望了解哪种形状最有利于产生超导体，而不因电阻而损失。相关的数学工具来自于变分法和非线性偏微分方程理论，以及应用于前面的渐近分析方法

In this project, the principal investigator pursues a research program in nonlinear partial differential equations, the calculus of variations and singular perturbation theory. The researchtopics come from three applied settings: Ginzburg-Landau type models for superconductivity, von Karman type models for thin film blistering, and a model in micromagnetics. In the area of superconductivity, the P.I. will focus on the response of samples to large magnetic fields, with particular attention paid to the bifurcation from the normal state to a superconducting state. In the area of thin film blisters, the P.I. will investigate the nature of instabilities of the blistered region through the analysis of various dynamical models for blister growth and thin film growth. Finally, in the area of micromagnetics, the P.I. will analytically explore a model thought to capture a new kind of magnetic wall structure associated with a geometric constriction within the sample. This project concerns the behavior of various materials when subjected to outside fields or when forced to take on specific shapes. The energy of these systems is generally described through a function, often called an `order parameter,' whose values indicate what state is taken on by the material under a given set of circumstances (such as geometry, applied fields,etc.). Through this type of study, one hopes to gain an understanding of what shapes are optimal for a given sample in order to enhance or diminish various physical effects. For example, in the case of a superconductor, one hopes to learn which shapes are most conducive to producing a supercurrent that conducts without losses due to resistance. The relevant mathematical tools come from the calculus of variations and from the theory of nonlinear partial differential equations, as well as from methods of asymptotic analysis as applied to the previou